CM-field explained

In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field.

The abbreviation "CM" was introduced by .

Formal definition

A number field K is a CM-field if it is a quadratic extension K/F where the base field F is totally real but K is totally imaginary. I.e., every embedding of F into

C lies entirely within R , but there is no embedding of K into R .

In other words, there is a subfield F of K such that K is generated over F by a single square root of an element, sayβ =

\sqrt{\alpha} ,in such a way that the minimal polynomial of β over the rational number field Q has all its roots non-real complex numbers. For this α should be chosen totally negative, so that for each embedding σ of F into the real number field,σ(α) < 0.

Properties

One feature of a CM-field is that complex conjugation on

C induces an automorphism on the field which is independent of its embedding into C . In the notation given, it must change the sign of β.

A number field K is a CM-field if and only if it has a "units defect", i.e. if it contains a proper subfield F whose unit group has the same

Z -rank as that of K . In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem.

Examples

• The simplest, and motivating, example of a CM-field is an imaginary quadratic field, for which the totally real subfield is just the field of rationals.

Q (\zeta _{n)} , which is generated by a primitive nth root of unity. It is a totally imaginary quadratic extension of the totally real field Q (\zeta _{n} + \zeta ^{-1} _{n} ). The latter is the fixed field of complex conjugation, and Q (\zeta _{n)} is obtained from it by adjoining a square root of \zeta ^{-2} _{n} -2 = (\zeta _{n} - \zeta ^{-1} _{n} ) ^{2.}

• The union Q^CM of all CM fields is similar to a CM field except that it has infinite degree. It is a quadratic extension of the union of all totally real fields Q^R. The absolute Galois group Gal(/Q^R) is generated (as a closed subgroup) by all elements of order 2 in Gal(/Q), and Gal(/Q^CM) is a subgroup of index 2. The Galois group Gal(Q^CM/Q) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(Q^R/Q).
• If V is a complex abelian variety of dimension n, then any abelian algebra F of endomorphisms of V has rank at most 2n over Z. If it has rank 2n and V is simple then F is an order in a CM-field. Conversely any CM field arises like this from some simple complex abelian variety, unique up to isogeny.
• One example of a totally imaginary field which is not CM is the number field defined by the polynomial

x ^{4} + x ^{3} - x ^{2} - x + 1 .

References

• • • • Book: Washington, Lawrence C.. Introduction to Cyclotomic fields. Springer-Verlag. New York. 1996. 2nd. 0-387-94762-0. 0966.11047.